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Circular Measurement of Angle

Prove that: s...

Question

Prove that: $\frac{sin 5 x + sin 3 x}{cos 5 x + cos 3 x} = tan 4 x$

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Solution

$\frac{sin 5 x + sin 3 x}{cos 5 x + cos 3 x} = tan 4 x$
$L . H . S = \frac{sin 5 x + sin 3 x}{cos 5 x + cos 3 x}$
$cos A + cos B = 2 cos (\frac{A + B}{2}) cos (\frac{A - B}{2})$
$sin A + sin B = 2 sin (\frac{A + B}{2}) cos (\frac{A - B}{2})$
$= \frac{2 sin (\frac{5 x + 3 x}{2}) cos (\frac{5 x - 3 x}{2})}{2 cos (\frac{5 x + 3 x}{2}) cos (\frac{5 x - 3 x}{2})} = \frac{2 sin (4 x) cos (x)}{2 cos (4 x) cos (x)}$
$= tan 4 x = R . H . S$

Hence proved.

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Circular Measurement of Angle

Standard XII Mathematics

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